let Y be non empty set ; for G being Subset of (PARTITIONS Y)
for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds
CompF (F,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J
let G be Subset of (PARTITIONS Y); for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds
CompF (F,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J
let A, B, C, D, E, F, J be a_partition of Y; ( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies CompF (F,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J )
{A,B,C,D,E,F,J} =
{A,B,C,D} \/ {E,F,J}
by ENUMSET1:19
.=
{A,B,C,D} \/ {F,E,J}
by ENUMSET1:58
.=
{A,B,C,D,F,E,J}
by ENUMSET1:19
;
hence
( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies CompF (F,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J )
by Th46; verum